Water as a “glue”: Elasticity-enhanced wet attachment of biomimetic microcup structures

Octopus, clingfish, and larva use soft cups to attach to surfaces under water. Recently, various bioinspired cups have been engineered. However, the mechanisms of their attachment and detachment remain elusive. Using a novel microcup, fabricated by two-photon lithography, coupled with in situ pressure sensor and observation cameras, we reveal the detailed nature of its attachment/detachment under water. It involves elasticity-enhanced hydrodynamics generating “self-sealing” and high suction at the cup-substrate interface, converting water into “glue.” Detachment is mediated by seal breaking. Three distinct mechanisms of breaking are identified, including elastic buckling of the cup rim. A mathematical model describes the interplay between the attachment/detachment process, geometry, elasto-hydrodynamics, and cup retraction speed. If the speed is too slow, then the octopus cannot attach; if the tide is too gentle for the larva, then water cannot serve as a glue. The concept of “water glue” can innovate underwater transport and manufacturing strategies.

Before the tests, the center of the cup was aligned with the center of the opening of the micro sensor. This was achieved by two step motors (VT-80 linear stages, PI, Karlsruhe, Germany) and visualization with a camera. For the adhesion tests, the cup was approached to the sensor with a velocity of 10 μm/s until a preload of 10 mN was reached. After 5 seconds of contact, the cup was retracted to move upward at a desire velocity (1~100 μm/s in the experiments) until a pull-off occurred. Here the maximum tensile load was defined as the pull-off force. The pull-off stress can be obtained by dividing the force with the projected area of the cup in the original undeformed state. Figure S3 gives the in-situ images with the membrane position at different retraction times as well as the deflection of force sensing membrane and the corresponding liquid pressures at different time points of retraction at the retraction velocity of ! = 10 µm/s.

S4. Analytical model of underwater suction cup and its parameter estimation
We develop a simple mathematical model of the cup with the sensor (Figure 2f). The model has three chambers, 1-3, with spring constants " , # , and $ . Chamber 1, representing the cup, is axi-symmetric with cross sectional area " .
It has a lip with width that comes in contact with a substrate. The contact is frictionless. A vertical spring with stiffness, ! , represents the stalk. Let ! be the applied prescribed retraction at the top end of the stalk, and ! is the corresponding measured force. Let " be the corresponding displacement at the base of the stalk. " accounts for the elastic deformation of the cup, as well as inward sliding and deformation of the lip. We employ a simple model with %& ! %' " = " , the spring constant of Chamber 1 (Figure 2e and 2f). Note that there is no fluid involved in defining " , and hence there is no pressure change in the chamber due to " .
The effective stiffness of the cup increases in the presence of liquid, herein considered incompressible. If liquid is not allowed to seep into the cup through the space between the lip and the substrate, then the deformation of the cup must satisfy volume conservation. Thus, during retraction, the cup cannot deform as freely, and its effective stiffness increases. Chamber 2 in the model accounts for this increased stiffness. Chambers 1 and 2 are connected, and hence they share the same suction pressure, . When the stalk is retracted, volume conservation requires " " = # # , where # and # are the cross-sectional area and the deformation of chamber 2. Force balance requires # = # # . Now, force, ! , required to move the stalk by ! is ! = ! ( ! − " ). " is obtained from force balance ! ( ! − " ) = " " + " . Then ! is given by Chamber 3 represents the pressure sensor. It has a cross sectional area of $ and spring constant, $ . It is connected to chambers 1 and 2, and hence shares the same suction pressure, with $ = $ $ . If liquid is not allowed to seep in from outside during retraction, then mass conservation requires " " = # # + $ $ . Allowing the flow, let be the flow per unit time into the suction cup as the stalk is retracted. Then mass balance requires: For an axisymmetric case, can be derived using Navier-Stokes equation and the assumption of steady state condition. is given by Here 567 and 8; are the water pressures outside and inside the suction cup, = 567 − 8 is the suction pressure in the cup, = +9 #

4/
, is the viscosity and 8 is the inner radius of the lip. Eq.(S1)-(S3) with force balance conditions, $ = $ $ and # = # # , give the evolution equation for pressure as the stalk is retracted with a prescribed velocity, !: We note that suction pressure develops between the lip and the substrate over the width, , of the lip (Figure 2g).
Suction reaches the value of at the inner periphery of the lip. Suction vanishes at the outer periphery of the lip. The lip is thus pulled towards the substrate by the suction. Both the lip and the substrate have asperities. As suction increases, the highest peaks first come in contact and deform elastically while smaller peaks come closer and eventually begin to contact. The effective gap, , decreases with increasing , but with increasing resistance. As shown in Figure 2g, the net force on the lip is given by =8-= < *#+ - ; is the area of the lip, and > is the force of the spring from chamber 1. Higher the net force, =8-, smaller is and higher is the resistance to further decrease in . We thus model versus =8- Here ! is the gap when =8-= 0, and * is a reference =8-at which Mass balance (Eq. S4) then gives: Here, = validity of such simple models is verified by comparing the trends of the system behavior predicted by the model with those observed experimentally in the physical system. We follow a similar strategy below. Experimentally, the measured quantities are, ! ( ), ! ( ), and chamber pressure, ( ), recorded by the pressure sensor. In the following, we study the model using the experimental cup tested in Figure 2 and Figure S2, with dimensions as follows: stalk radius, 8 = 80 , stalk length, >7I=J = 250 , lip radius, " = 120 , lip thickness, = 10 , elastic modulus of cup material, ≈ 10 , Poisson's ratio, = .41. We will estimate the parameters of the model corresponding to the physical cup, as well as compare model predictions with experimental observations. We determine the model parameters, ! , , " , " and for the cup firstly with ! = 10 µm/s.
Determination of " and " : Eq. (S1) gives the time derivative of the force, ̇! , as the stalk is retracted: where ! is the stiffness of the stalk given by ! = K< /&4*5 Here, is the elastic modulus, >7I=J the crosssectional area of the stalk, and >7I=J the stalk length. Using the data shown in Figure 2d and Figure S3b for the retraction velocity, ! = 10 µm/s, we plot ̇! versus ̇ for the duration of 6 s (Figures 2h and S4a). The model (Eq. (S10)) predicts a linear relation between them. Experiment verifies the linearity. The slope of the linear fit gives ! " ( ! + " ) and the intercept gives . Thus, " = 31361 µm # and " = 0.0579 mN/µm.
The speed dependence of may originate from viscoelasticity of cup material, polyurethanes. The model considers the material as linear elastic. In order to correct this discrepancy, we apply a correction factor for the time constant, as Q (velocity dependent time constant) as Taking log on both sides, ln( Q ) = ln( ) − ln(1 +̇!), and linearly fitting the data in the log-log plot ( Figure S4e), we can get: = 0.92. Note that ! in Eq. (S14) can be considered as normalized by 1 μm/s, i.e., ! is nondimensional. Figure S4e further shows that the variation in the rest of the parameters ( ! , , " , " ) is small even when the retraction velocity changes by two orders of magnitude. The similarity between the predicted and experimental trends in cup response discussed above suggests that the model represents the suction cups in liquids.
To further test the model, we conduct retraction experiment with ! = 5, and 20 µm/s and measure the corresponding force of retraction until detachment. We also apply the model to predict the force-time history using the parameters. The model predictions almost coincide with experimental observation (Figure 2j, Figure S4f), except near detachment or failure, which we discuss next.

S5. Failure of the cup by buckling: finite element analysis
In order to test whether the lip of the cup can buckle during retraction, we simulate the deformation of the lip using finite element analysis (Figure 4d). The lip geometry and material properties are similar to those in the experiment (Fig. 4c, outer lip radius = 60 µm, stalk radius = 40 µm, thickness = 5 µm, elastic modulus = 10 MPa, Poisson's ratio = 0.41). However, the lip is treated as an independent plate isolated from the stalk. The analysis consists of two stages: (1) Eigen analysis of the lip to determine the lowest buckled mode shape. The inner perimeter of the lip is constrained from out of plane displacement, but it is subjected to an inward radial displacement. The rest of the lip is free. At a critical inward displacement, the lip buckles. This gives the mode shape of the lip at buckling (Figure 4d-i).
(2) In order to simulate the experimental condition of retraction, we now model the lip on a substrate, where lipsubstrate contact is frictionless. The lip can deform out of plane away from the substrate. In order to mimic suction as in case of the experiments, an out-of-plane pressure is applied on the plate that decreases radially outward. The pressure is highest (1 MPa) at the inner perimeter, and it vanishes at the outer perimeter ( Fig. 4d-i). The pressure pushes the lip towards the substrate. In order to test potential buckling, we apply small point loads (2 µN) at discrete locations along the outer perimeter. These locations are the peaks (2 µN pointing upward) and valleys (2 µN pointing downward) of the free buckled plate (Fig. 4d-i). These small loads induce "imperfections" in the plate, and they allow to carry out post buckling analysis. The inner perimeter of the lip is restricted to in-plane displacement only. It is subjected to radially inward displacement, mimicking radial motion during retraction. If buckling is energetically favorable under these constraints, then we expect a non-linear relation between the radial displacement and out of plane deformation along the point loads. The amplitude, S6RJ=D , of the out-of-plane deformation of the lip at the locations of 2 µN load increases with the radial motion of the outer perimeter. S6RJ=D is expected to increase rapidly near a critical RT (Figure 4d-iii). Without any imperfection, S6RJ=D is expected to increase as √ when > RT . We thus fit a √ curve to the simulation results of vs S6RJ=D . The curve meets the -axis at RT ≈ 4 µm when the circumferential strain is about 6.7%. We consider this strain as the critical buckling strain for the underwater cup. This strain cannot be detected experimentally. Once buckling instability initiates, the amplitude increases rapidly with retraction. The experimental image prior to detachment shows (Figure 4c) that the outer diameter of the lip is 52 µm, implying a strain of 13.5%. Simulation also shows buckling amplitude of about 14 µm at a strain of about 13.3%.

S6. Effect of scaling on attachment strength
We derive the scaling law from the analytical model described above. We consider two cases for scaling: ! ∼ ( ) " , and ! ∼ ( ) ! . The former represents aquatic animals in natural biological contacts, and the latter reflects our experimental setup where the resolution of 3D printing of the cups or the roughness of the substrate do not change with cup size. In our scaling analysis, we assume = ! in Eq. S5, i.e., the gap between the lip and the substrate does not change with suction. Let represent the size scale of the cup.
To determine the scale dependence of strength, we note that the cup may detach in three independent modes during retraction. Mode I: The lip does not get pulled-in towards the substrate and suction does not develop. Here, detachment occurs as soon as the stalk is retracted. This occurs if the rate of increase of retraction force exceeds the rate of increase of suction force at time, = 0. This mode is avoided by increasing the rate of increase of suction force for a given rate of retraction, !, i.e., by increasing the lip width such that < *#+ #< > " / # . Once the lip is engaged, then the cup may detach either by detaching when the retraction force exceeds the suction force (Mode II), or by buckling the lip out of plane (Mode III). Mode II detachment occurs when =8-= 0 (Eq S6). This gives Here, % is the time to detachment in Mode II. % can be solved from Eq. S17 for a given = . ~* $ when !~! ; and ~! when !~" . From Eq. S17, 7 7 E , %~! . Thus, %~$ when !~! , and %~! when !~" . At the impending detachment (Mode II, =8-= < *#+ -# − " " = 0), attachment force, % is given by Eqs. S1, S15 and S17: Then, attachment strength, YI' ~ $ *" ! ~ # when !~! , i.e., strength decreases with size, as expected, since a fixed ! would appear large as cup size becomes very small. The cup will offer little adhesion due to fluid entrée to the cup through the relatively large gap, ! . But YI' ~ ! *" ! ~ *" when !~" , i.e., smaller cups would become stronger. If decreases with increasing suction, then the strength is espected to be even higher. Softer the lip, higher is its deformability, and more conformal it will be with the substrate with suction. This will decrease the gap further and increase the cup strength.
At buckling (Mode III), Eq. S1 gives the detachment force Here !S is the displacement of the stalk at the time, S , when buckling initiates. If S is the circumferential compressive strain of the lip (outer rim) at buckling, then !S =̇! S ≈ S ( 8 + ) + >7I=J >7I=J . Here, >7I=J is the elastic strain of the stalk of length >7I=J , S is the buckling strain (circumferential) at the lip periphery, given by, S = Δ( 8 + )/( 8 + ). The approximation above is because a fraction of Δ( 8 + ) contributes to !S . Since both >7I=J and S are the elastic strains induced by S , >7I=J and S are proportional to each other. Since S is the threshold strain for buckling, it is scale independent, just as buckling strain of Euler column is independent of its size.
Then, >7I=J at buckling is also size independent. Thus, !S~" and S = ' !8 for a given !. It follows that larger the cup size, the longer it takes to initiate buckling.
From Eqs. S15 and S19, detachment force at buckling, S , is S20, YI' S~! + !~! for small cups, and ~* " + !~! for large cups. This implies that attachment strength in buckling mode is independent of cup size for a prescribed !.
Cups can be designed with specific desired failure mode. For example, one can design a cup that reaches failure by detaching in Mode II when % = . Eqs. S15 and S17 then give the necessary geometric condition: Here, *" ≈ 1/3 is used. The latter inequality in Eq. S21 prevents failure by Mode I. To ensure that buckling does not occur before Mode II debonding, we require S > , which gives For a given retraction speed of !, the above condition can be met by reducing lip width , and 8 , and by tuning the stiffnesses, 8 .
Recall that time to buckling S~, and time to de-bond in mode II, %~$ (when !~! , our experimental case). Thus, as the cup size increases, buckling mode is expected to precede the debonding mode. When !~" , then %~! while S remains as S~. Hence, debonding failure by Mode II is likely to occur with increasing size. For a given cup size, S increases with decreasing retraction speed, ! as S~1 /̇!, whereas % is independent of !. Thus, for a given cup, the detachment mode changes with increasing ! from Mode II to Mode III.

Experimental verification of the scaling law:
We fabricated 6 cups (design shown in Fig 1b)  and their attachment strength remained nearly constant with change in size by nearly an order of magnitude, but strength increased with increase in elastic modulus, E, of cups (Fig. S5).
We derive three insights from these observations in relation to our model and scaling laws.
(1) Our model (Eqs. S20) predicts that the pull-off strength is size independent in Mode III, YI' S~8! . This is observed in the experiments when 8 increased from 40 to 80 µm. increase in attachment strength by more than 2 times for all the cups tested when their E increased from 4 to 10 MPa.   in-situ images to monitor the membrane position at different retraction times. The initial distance between the membrane tongue and fixed reference tongue is 5.5 μm. Additional deflection due to suction is listed in (b). Image analysis was conducted with ImageJ.